Question

\(\int \frac{dx}{25 - x^2} =\)

• A. \(\frac{1}{5} \log \left| \frac{x-5}{x+5} \right| + c\)
• B. \(\frac{1}{5} \log \left| \frac{x+5}{x-5} \right| + c\)
• C. \(\frac{1}{10} \log \left| \frac{5+x}{5-x} \right| + c\)
• D. \(\frac{1}{10} \log \left| \frac{5-x}{5+x} \right| + c\)

Hint:

Memorizing the standard integral forms for \(\frac{1}{a^2-x^2}\), \(\frac{1}{x^2-a^2}\), and \(\frac{1}{a^2+x^2}\) is essential for speed and accuracy in competitive exams. They appear very frequently.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Evaluate ∫₀¹ x(1-x)⁹ dx.

Step 2: Key Formula (Alternate):
Use King property: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx.

Step 3: Detailed Explanation:
I = ∫₀¹ x(1-x)⁹ dx = ∫₀¹ (1-x)x⁹ dx = ∫₀¹ (x⁹-x¹⁰)dx = [x¹⁰/10 - x¹¹/11]₀¹ = 1/10-1/11 = 1/110.

Step 4: Final Answer:
Value is 1/110.